Abstract
In this paper, we give or improve compression-expansion results for set contractions in conical domains determined by balls or star convex sets. In the compression case, we use Potter’s idea of proof, while the expansion case is reduced to the compression one by means of a change of variable. Finally, to illustrate the theory, we give an application to the initial value problem for a system of implicit first order differential equations.
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Acknowledgements
The authors thank the anonymous referee for suggesting to give the application more generally to systems than to a single equation. Also for giving us the idea to apply the results to boundary value problems for PDEs, a theme for possible future research. Cristina Lois-Prados and Rosana Rodríguez-López acknowledge the support of the research grant MTM2016-75140-P (AEI/FEDER, UE). The research of Cristina Lois-Prados has been partially supported by grant ED481A-2018/080 from Xunta de Galicia. The authors also thank Daniel Cao Labora for his idea of the proof of Theorem 2.2.
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Lois-Prados, C., Precup, R. & Rodríguez-López, R. Krasnosel’skii type compression-expansion fixed point theorem for set contractions and star convex sets. J. Fixed Point Theory Appl. 22, 63 (2020). https://doi.org/10.1007/s11784-020-00799-0
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DOI: https://doi.org/10.1007/s11784-020-00799-0